Theorem mulgnnp1 13581

Description: Group multiple (exponentiation) operation at a successor. (Contributed by Mario Carneiro, 11-Dec-2014.)

Hypotheses

Ref	Expression
mulg1.b	|- B = ( Base ` G )
mulg1.m	|- .x. = (.g ` G )
mulgnnp1.p	|- .+ = ( +g ` G )

Assertion

Ref	Expression
mulgnnp1	|- ( ( N e. NN /\ X e. B ) -> ( ( N + 1 ) .x. X ) = ( ( N .x. X ) .+ X ) )

Proof of Theorem mulgnnp1

Dummy variables u v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 109	. . . . 5 |- ( ( N e. NN /\ X e. B ) -> N e. NN )
2		nnuz 9719	. . . . 5 |- NN = ( ZZ>= ` 1 )
3	1, 2	eleqtrdi 2300	. . . 4 |- ( ( N e. NN /\ X e. B ) -> N e. ( ZZ>= ` 1 ) )
4		simplr 528	. . . . 5 |- ( ( ( N e. NN /\ X e. B ) /\ u e. ( ZZ>= ` 1 ) ) -> X e. B )
5		simpr 110	. . . . . 6 |- ( ( ( N e. NN /\ X e. B ) /\ u e. ( ZZ>= ` 1 ) ) -> u e. ( ZZ>= ` 1 ) )
6	5, 2	eleqtrrdi 2301	. . . . 5 |- ( ( ( N e. NN /\ X e. B ) /\ u e. ( ZZ>= ` 1 ) ) -> u e. NN )
7		fvconst2g 5821	. . . . . . 7 |- ( ( X e. B /\ u e. NN ) -> ( ( NN X. { X } ) ` u ) = X )
8		simpl 109	. . . . . . 7 |- ( ( X e. B /\ u e. NN ) -> X e. B )
9	7, 8	eqeltrd 2284	. . . . . 6 |- ( ( X e. B /\ u e. NN ) -> ( ( NN X. { X } ) ` u ) e. B )
10	9	elexd 2790	. . . . 5 |- ( ( X e. B /\ u e. NN ) -> ( ( NN X. { X } ) ` u ) e. _V )
11	4, 6, 10	syl2anc 411	. . . 4 |- ( ( ( N e. NN /\ X e. B ) /\ u e. ( ZZ>= ` 1 ) ) -> ( ( NN X. { X } ) ` u ) e. _V )
12		simprl 529	. . . . 5 |- ( ( ( N e. NN /\ X e. B ) /\ ( u e. _V /\ v e. _V ) ) -> u e. _V )
13		mulg1.b	. . . . . . . 8 |- B = ( Base ` G )
14	13	basmex 13006	. . . . . . 7 |- ( X e. B -> G e. _V )
15		mulgnnp1.p	. . . . . . . 8 |- .+ = ( +g ` G )
16		plusgslid 13059	. . . . . . . . 9 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
17	16	slotex 12974	. . . . . . . 8 |- ( G e. _V -> ( +g ` G ) e. _V )
18	15, 17	eqeltrid 2294	. . . . . . 7 |- ( G e. _V -> .+ e. _V )
19	14, 18	syl 14	. . . . . 6 |- ( X e. B -> .+ e. _V )
20	19	ad2antlr 489	. . . . 5 |- ( ( ( N e. NN /\ X e. B ) /\ ( u e. _V /\ v e. _V ) ) -> .+ e. _V )
21		simprr 531	. . . . 5 |- ( ( ( N e. NN /\ X e. B ) /\ ( u e. _V /\ v e. _V ) ) -> v e. _V )
22		ovexg 6001	. . . . 5 |- ( ( u e. _V /\ .+ e. _V /\ v e. _V ) -> ( u .+ v ) e. _V )
23	12, 20, 21, 22	syl3anc 1250	. . . 4 |- ( ( ( N e. NN /\ X e. B ) /\ ( u e. _V /\ v e. _V ) ) -> ( u .+ v ) e. _V )
24	3, 11, 23	seq3p1 10647	. . 3 |- ( ( N e. NN /\ X e. B ) -> ( seq 1 ( .+ , ( NN X. { X } ) ) ` ( N + 1 ) ) = ( ( seq 1 ( .+ , ( NN X. { X } ) ) ` N ) .+ ( ( NN X. { X } ) ` ( N + 1 ) ) ) )
25		id 19	. . . . 5 |- ( X e. B -> X e. B )
26		peano2nn 9083	. . . . 5 |- ( N e. NN -> ( N + 1 ) e. NN )
27		fvconst2g 5821	. . . . 5 |- ( ( X e. B /\ ( N + 1 ) e. NN ) -> ( ( NN X. { X } ) ` ( N + 1 ) ) = X )
28	25, 26, 27	syl2anr 290	. . . 4 |- ( ( N e. NN /\ X e. B ) -> ( ( NN X. { X } ) ` ( N + 1 ) ) = X )
29	28	oveq2d 5983	. . 3 |- ( ( N e. NN /\ X e. B ) -> ( ( seq 1 ( .+ , ( NN X. { X } ) ) ` N ) .+ ( ( NN X. { X } ) ` ( N + 1 ) ) ) = ( ( seq 1 ( .+ , ( NN X. { X } ) ) ` N ) .+ X ) )
30	24, 29	eqtrd 2240	. 2 |- ( ( N e. NN /\ X e. B ) -> ( seq 1 ( .+ , ( NN X. { X } ) ) ` ( N + 1 ) ) = ( ( seq 1 ( .+ , ( NN X. { X } ) ) ` N ) .+ X ) )
31		mulg1.m	. . . 4 |- .x. = (.g ` G )
32		eqid 2207	. . . 4 |- seq 1 ( .+ , ( NN X. { X } ) ) = seq 1 ( .+ , ( NN X. { X } ) )
33	13, 15, 31, 32	mulgnn 13577	. . 3 |- ( ( ( N + 1 ) e. NN /\ X e. B ) -> ( ( N + 1 ) .x. X ) = ( seq 1 ( .+ , ( NN X. { X } ) ) ` ( N + 1 ) ) )
34	26, 33	sylan 283	. 2 |- ( ( N e. NN /\ X e. B ) -> ( ( N + 1 ) .x. X ) = ( seq 1 ( .+ , ( NN X. { X } ) ) ` ( N + 1 ) ) )
35	13, 15, 31, 32	mulgnn 13577	. . 3 |- ( ( N e. NN /\ X e. B ) -> ( N .x. X ) = ( seq 1 ( .+ , ( NN X. { X } ) ) ` N ) )
36	35	oveq1d 5982	. 2 |- ( ( N e. NN /\ X e. B ) -> ( ( N .x. X ) .+ X ) = ( ( seq 1 ( .+ , ( NN X. { X } ) ) ` N ) .+ X ) )
37	30, 34, 36	3eqtr4d 2250	1 |- ( ( N e. NN /\ X e. B ) -> ( ( N + 1 ) .x. X ) = ( ( N .x. X ) .+ X ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2178   _Vcvv 2776   {csn 3643   X. cxp 4691   ` cfv 5290  (class class class)co 5967   1c1 7961   + caddc 7963   NNcn 9071   ZZ>=cuz 9683   seqcseq 10629   Basecbs 12947   +g cplusg 13024  ._gcmg 13570

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-inn 9072  df-2 9130  df-n0 9331  df-z 9408  df-uz 9684  df-seqfrec 10630  df-ndx 12950  df-slot 12951  df-base 12953  df-plusg 13037  df-0g 13205  df-minusg 13451  df-mulg 13571

This theorem is referenced by:  mulg2  13582  mulgnn0p1  13584  mulgnnass  13608  gsumfzconst  13792